Order Of Convergence Research Articles

A Globally Convergent Iterative Method for Matrix Sign Function and Its Application for Determining the Eigenvalues of a Matrix Pencil

In this research article, we propose a new matrix iterative method with a convergence order of five for computing the sign of a complex matrix by examining the different patterns and symmetry of existing methods. Analysis of the convergence of the method was explored on a global scale, and attraction basins were demonstrated. In addition to this, the asymptotic stability of the scheme was explored.Then, an algorithm for determing thegeneralized eigenvalues for the case of regular matrix pencils was investigated using the matrix sign computation. We performed a series of numerical experiments using numerous matrices to confirm the usefulness and superiority of the proposed method.

HDG Method for Nonlinear Parabolic Integro-Differential Equations

Abstract The hybridizable discontinuous Galerkin (HDG) method has been applied to a nonlinear parabolic integro-differential equation. The nonlinear functions are considered to be Lipschitz continuous to analyze uniform in time a priori bounds. An extended type Ritz–Volterra projection is introduced and used along with the HDG projection as an intermediate projection to achieve optimal order convergence of O ⁢ ( h k + 1 ) O(h^{k+1}) when polynomials of degree k ≥ 0 k\geq 0 are used to approximate both the solution and the flux variables. By relaxing the assumptions in the nonlinear variable, super-convergence is achieved by element-by-element post-processing. Using the backward Euler method in temporal direction and quadrature rule to discretize the integral term, a fully discrete scheme is derived along with its error estimates. Finally, with the help of numerical examples in two-dimensional domains, computational results are obtained, which verify our results.

An efficient iterative method for matrix sign function with application in stability analysis of control systems using spectrum splitting

The goal of this study is to construct a novel iterative method to compute the matrix sign function using a different approach. It is discussed that the new method is globally convergent and asymptotically stable. It achieves the sixth order of convergence and only requires five matrix–matrix multiplications. The obtained results are extended to compute the number of eigenvalues of a matrix in a specified region of the complex plane. This is done by performing appropriate sequence of matrix sign computations. An application of this technique has been discussed in stability analysis of linear time‐invariant dynamic systems in control theory. Numerical results have been given to justify the effectual performance and superiority of the proposed method. Matrices of various sizes have been considered for this purpose.

Global convergence analysis of Caputo fractional Whittaker method with real world applications

The present article deals with the effect of convexity in the study of the well-known Whittaker iterative method, because an iterative method converges to a unique solution \(t^*\) of the nonlinear equation \(\psi(t)=0\) faster when the function's convexity is smaller. Indeed, fractional iterative methods are a simple way to learn more about the dynamic properties of iterative methods, i.e., for an initial guess, the sequence generated by the iterative method converges to a fixed point or diverges. Often, for a complex root search of nonlinear equations, the selective real initial guess fails to converge, which can be overcome by the fractional iterative methods. So, we have studied a Caputo fractional double convex acceleration Whittaker's method (CFDCAWM) of order at least (\(1+2\zeta\)) and its global convergence in broad ways. Also, the faster convergent CFDCAWM method provides better results than the existing Caputo fractional Newton method (CFNM), which has (\(1+\zeta\)) order of convergence. Moreover, we have applied both fractional methods to solve the nonlinear equations that arise from different real-life problems.

Modulated Lacunary 𝒫-Statistical Convergence of Order γ of Sequences of Fuzzy Functions

The central objective of this paper is to investigate the lacunary statistical convergence of order [Formula: see text] and strong lacunary summability of order [Formula: see text] for sequences of fuzzy functions by using the modulus function and a regular matrix. We examine the relevant inclusion relations between these spaces under specific conditions. Additionally, we study properties related to these spaces and establish several interesting results.

Efficient Fourth-Order Weights in Kernel-Type Methods without Increasing the Stencil Size with an Application in a Time-Dependent Fractional PDE Problem

An effective strategy to enhance the convergence order of nodal approximations in interpolation or PDE problems is to increase the size of the stencil, albeit at the cost of increased computational burden. In this study, our goal is to improve the convergence orders for approximating the first and second derivatives of sufficiently differentiable functions using the radial basis function-generated Hermite finite-difference (RBF-HFD) scheme. By utilizing only three equally spaced points in 1D, we are able to boost the convergence rate to four. Extensive tests have been conducted to demonstrate the effectiveness of the proposed theoretical weighting coefficients in solving interpolation and PDE problems.

A comparative study on the acceleration techniques for solving finite difference discretization poisson’s equation in the PIC/MCC Method

A wide variety of plasma phenomena have been investigated during the past decades using the particle-in-cell/Monte Carlo collisions (PIC/MCC) method. As an important component of the PIC/MCC method, solving Poisson’s equation is crucial for the accuracy and efficiency of calculations. Different acceleration techniques for solving finite difference discretization Poisson’s equation are investigated and compared, including direct method, iterative method, multigrid (MG) method, parallel computing and inherited initial value. The charge density distribution with a known analytical solution is used to validate the algorithm and code. The optimal relaxation factor for the successive over-relaxation (SOR) method in 2D Poisson’s equation with unequal grid node numbers in different dimensions is derived, which is only related to the dimension with the largest grid number. Although there will be a ‘more optimal’ relaxation factor deviated from in some simulation cases, selecting the optimal relaxation factor derived always leads to a not slow solving speed. However, when SOR is used in MG for smoothing, the optimal relaxation factor will shift to 0.5–1.2 from the theoretical optimal value derived with the increase of MG levels. By comparing the convergence order under different relaxation factors and MG levels, the suitable MG level is proposed as log2[min(N x , N y )]−2. Combining the optimal SOR relaxation factor, MG, parallel computing and inherited initial values, the computational cost may decrease by 5 orders of magnitude than that by the simple Gaussian elimination (GE). Based on the optimal acceleration techniques mentioned above, a benchmark simulation case electron cyclotron drift instability (ECDI) in magnetized plasmas was run to further validate the developed PIC/MCC code. The distributions of electric field in the x-direction, electron density and electron temperature are all consistent with the literatures. This paper provides a reference for the acceleration strategy selection for solving Poisson’s equation quickly in plasma simulations.

Monolithic and local time-stepping decoupled algorithms for transport problems in fractured porous media

Abstract The objective of this paper is to develop efficient numerical algorithms for the linear advection-diffusion equation in fractured porous media. A reduced fracture model is considered where the fractures are treated as interfaces between subdomains and the interactions between the fractures and the surrounding porous medium are taken into account. The model is discretized by a backward Euler upwind-mixed hybrid finite element method in which the flux variable represents both the advective and diffusive fluxes. The existence, uniqueness, as well as optimal error estimates in both space and time for the fully discrete coupled problem are established. Moreover, to facilitate different time steps in the fracture-interface and the subdomains, global-in-time, nonoverlapping domain decomposition is utilized to derive two implicit iterative solvers for the discrete problem. The first method is based on the time-dependent Steklov–Poincaré operator, while the second one employs the optimized Schwarz waveform relaxation (OSWR) approach with Ventcel-Robin transmission conditions. A discrete space-time interface system is formulated for each method and is solved iteratively with possibly variable time step sizes. The convergence of the OSWR-based method with conforming time grids is also proved. Finally, numerical results in two dimensions are presented to verify the optimal order of convergence of the monolithic solver and to illustrate the performance of the two decoupled schemes with local time-stepping on problems of high Péclet numbers.

Reducing the Constrained Multibody Dynamics Problem to the Solution of a System of ODEs via Velocity Partitioning and Lie Group Integration

Abstract In multibody dynamics, formulating the equations of motion in absolute Cartesian coordinates results in a set of index-3 differential algebraic equations (DAEs). In this work, we present an approach that bypasses the DAE problem by partitioning the velocities in the system into dependent and independent coordinates, thereby reducing the task of producing the time evolution of the mechanical system to one of solving a set of ordinary differential equations (ODEs). In this approach, the independent coordinates are integrated directly, while the dependent coordinates are recovered through the kinematic constraint equations at the position and velocity levels. Notably, Lie group integration is employed to directly obtain the orientation matrix A at each time step of the simulation. This eliminates the need to choose generalized coordinates to capture the orientation of a body, as the matrix A is a byproduct of the solution algorithm. The approach is akin to the method presented in [1], which combines coordinate partitioning with an Euler parameter formulation. Herein, we outline the new approach and demonstrate it in conjunction with four mechanisms: a single pendulum, a double pendulum, a four-link mechanism, and a slider crank. We report on the convergence order behavior of the proposed method and a performance comparison with the solution introduced in [1]. The Python code developed to generate the reported results is open-source and available in a public repository for reproducibility studies [2].

Linearly-fitted energy-mass-preserving schemes for Korteweg–de Vries equations

In this paper, second-order and fourth-order linearly-fitted energy-mass-preserving schemes for Korteweg–de Vries (KdV) equations are proposed. First, the KdV equation is semi-discretized into an oscillatory Hamiltonian system of Ordinary Differential Equations (ODEs) by semi-discretizations of the skew adjoint operator and the Hamiltonian in the Hamiltonian form of the KdV equation. Then the resulting semi-discrete system is integrated by exponential average vector field integrator. The order of convergence and the conservation properties are established for the introduced full-discrete procedures. It shows that the proposed methods are energy-preserving, mass-preserving and qualitatively preserve the dispersive relation well. Numerical experiments are carried out to validate the efficiency and the accuracy of the proposed schemes.

On the accuracy of shock-capturing schemes when calculating Cauchy problems with periodic discontinuous initial data

Abstract We study the accuracy of shock-capturing schemes for the shallow water Cauchy problems with piecewise smooth discontinuous initial data. We consider the second order balance-characteristic (CABARETM) scheme, the third order finite-difference Rusanov–Burstein–Mirin (RBM) scheme and the fifth order in space, the third order in time weighted essentially non-oscillatory (WENO5) scheme. We have shown that the maximum loss of accuracy occurs in the centered rarefaction waves of the exact solutions, where all these schemes have the first order of convergence and fairly close values of the numerical disbalances (errors), regardless of their formal approximation order on the smooth solutions. In the same time, inside the shock influence areas the considered schemes can have different convergence orders and, as a result, significantly different accuracy. In particular, when solving the Cauchy problem with periodic initial data, when the exact solution has no centered rarefaction waves, the RBM scheme has a significantly higher accuracy inside the shock influence areas, compared to the CABARETM and WENO5 schemes. It means that the combined scheme, in which the RBM scheme is a basic scheme and the CABARETM scheme is an internal one, can be effectively used to compute weak solutions of such type Cauchy problems.

A Crank–Nicolson leap-frog scheme for the unsteady incompressible magnetohydrodynamics equations

This paper presents a Crank–Nicolson leap-frog (CNLF) scheme for the unsteady incompressible magnetohydrodynamics (MHD) equations. The spatial discretization adopts the Galerkin finite element method (FEM), and the temporal discretization employs the CNLF method for linear terms and the semi-implicit method for nonlinear terms. The first step uses Stokes style’s scheme, the second step employs the Crank–Nicolson extrapolation scheme, and others apply the CNLF scheme. We establish that the fully discrete scheme is stable and convergent when the time step is less than or equal to a positive constant. Firstly, we show the stability of the scheme by means of the mathematical induction method. Next, we focus on analyzing error estimates of the CNLF method, where the convergence order of the velocity and magnetic field reach second-order accuracy, and the pressure is the first-order convergence accuracy. Finally, the numerical examples demonstrate the optimal error estimates of the proposed algorithm.

Updating to Optimal Parametric Values by Memory-Dependent Methods: Iterative Schemes of Fractional Type for Solving Nonlinear Equations

In the paper, two nonlinear variants of the Newton method are developed for solving nonlinear equations. The derivative-free nonlinear fractional type of the one-step iterative scheme of a fourth-order convergence contains three parameters, whose optimal values are obtained by a memory-dependent updating method. Then, as the extensions of a one-step linear fractional type method, we explore the fractional types of two- and three-step iterative schemes, which possess sixth- and twelfth-order convergences when the parameters’ values are optimal; the efficiency indexes are 6 and 123, respectively. An extra variable is supplemented into the second-degree Newton polynomial for the data interpolation of the two-step iterative scheme of fractional type, and a relaxation factor is accelerated by the memory-dependent method. Three memory-dependent updating methods are developed in the three-step iterative schemes of linear fractional type, whose performances are greatly strengthened. In the three-step iterative scheme, when the first step involves using the nonlinear fractional type model, the order of convergence is raised to sixteen. The efficiency index also increases to 163, and a third-degree Newton polynomial is taken to update the values of optimal parameters.

Non-differentiable kernel-based approximation of memory-dependent derivative for drug delivery applications

Although memory-dependent derivative (MDD) has recently been the subject of extensive study, only one numerical approximation has been reported in the literature. Hence, this study introduces a novel approximation for MDD. Moreover, a new form of the kernel function is presented. The convergence order of our approximation is Oh3−S,0<S<1 , where (1−S) is the exponent of the kernel function. The proposed approach is used to numerically solve the memory-dependent advection-diffusion problem, and the numerical scheme's stability and convergence are discussed. Also, memory-based models have been described to study drug delivery and its diffusion from multi-layer capsules/tablets. These models are based on the fractional derivative and the MDD to solve the paradox of the unphysical feature of the infinite propagation speed of the published Fickian-based models. The theoretical analysis is validated with numerical examples by investigating the convergence order that is 3−S in time. To illustrate the validity and efficiency of the proposed models, profiles of concentration and drug mass are compared with the observed in vivo data. It is observed to be highly accurate and compatible with the in vivo data when compared with Fick's model.

Numerical Simulation for Solitary Waves of the Generalized Zakharov Equation Based on the Lattice Boltzmann Method

The generalized Zakharov equation is a widely used and crucial model in plasma physics, which helps to understand wave particle interactions and nonlinear wave propagation in plasma. The solitary wave solution of this equation provides insights into phenomena such as electron and ion acoustic waves, as well as magnetic field disturbances in plasma. The numerical simulation of solitary wave solutions to the generalized Zakharov equation is an interesting problem worth studying. This is crucial for plasma-based technology, as well as for understanding nonlinear wave propagation in plasma physics and other fields. In this study, a numerical investigation of the generalized Zakharov equation using the lattice Boltzmann method has been conducted. The lattice Boltzmann method is a new modeling and simulating method at the mesoscale. A lattice Boltzmann model was constructed by performing Taylor expansion, Chapman–Enskog expansion, and time multiscale expansion on the lattice Boltzmann equation. By defining the moments of the equilibrium distribution function appropriately, the macroscopic equation can be restored. Furthermore, the numerical experiments for the equation are carried out with the parameter lattice size m=100, time step Δt=0.001, and space step size Δx=0.4. The solitary wave solution of the equation is numerically simulated. Numerical results under different parameter values are compared, and the convergence and effectiveness of the model are numerically verified. It is obtained that the model is convergent in time and space, and the convergence orders are all 2.24881. The effectiveness of our model was also verified by comparing the numerical results of different numerical methods. The lattice Boltzmann method demonstrates advantages in both accuracy and CPU time. The results indicate that the lattice Boltzmann method is a good tool for computing the generalized Zakharov equation.

On the number of terms in the COS method for European option pricing

AbstractThe Fourier-cosine expansion (COS) method is used to price European options numerically in a very efficient way. To apply the COS method, one has to specify two parameters: a truncation range for the density of the log-returns and a number of terms N to approximate the truncated density by a cosine series. How to choose the truncation range is already known. Here, we are able to find an explicit and useful bound for N as well for pricing and for the sensitivities, i.e., the Greeks Delta and Gamma, provided the density of the log-returns is smooth. We further show that the COS method has an exponential order of convergence when the density is smooth and decays exponentially. However, when the density is smooth and has heavy tails, as in the Finite Moment Log Stable model, the COS method does not have exponential order of convergence. Numerical experiments confirm the theoretical results.

The Boyle–Romberg Trinomial Tree, a Highly Efficient Method for Double Barrier Option Pricing

Oscillations in option price convergence have long been a problematic aspect of tree methods, inhibiting the use of repeated Richardson extrapolation that could otherwise greatly accelerate convergence, a feature integral to some of the most efficient modern methods. These oscillations are typically caused by the fluctuating positions of nodes around the discontinuities in the payoff function or its derivatives. Our paper addresses this crucial gap that typically prohibits the use of lattice methods when high efficiency is needed. Focusing on double barrier options, we develop a trinomial tree in which the positions of the nodes are precisely adjusted to align with these discontinuities throughout the option’s lifespan and across various time steps. This alignment enables the use of repeated extrapolation to achieve high order convergence, including near barriers, a well-known challenge in many tree methods. Maintaining the inherent simplicity and adaptability of tree methods, our approach is easily applicable to other models and option types.

A locally calculable P3‐pressure in a decoupled method for incompressible Stokes equations

AbstractThis article will suggest a new finite element method to find a ‐velocity and a ‐pressure solving incompressible Stokes equations at low cost. The method solves first the decoupled equation for a ‐velocity. Then, using the calculated velocity, a locally calculable ‐pressure will be defined component‐wisely. The resulting ‐pressure is analyzed to have the optimal order of convergence. Since the pressure is calculated by local computation only, the chief time cost of the new method is on solving the decoupled equation for the ‐velocity. Besides, the method overcomes the problem of singular vertices or corners.

A Crank-Nicolson WG-FEM for unsteady 2D convection-diffusion equation with nonlinear reaction term on layer adapted mesh

This paper is contributed to explore how a Crank-Nicolson weak Galerkin finite element method (WG-FEM) addresses the singularly perturbed unsteady convection-diffusion equation with a nonlinear reaction term in 2D. The problem and some asymptotic behavior results are given for the exact solution and its derivatives with the parameter ε. These results are essential for proving the uniform convergence of the proposed WG-FEM. A tensor product Shishkin mesh featuring piece-wise discontinuous bilinear polynomials is employed to handle the layers for uniform convergence at different parameter values of ε. An error estimate ‖uN−INu‖WG is presented, where INu denotes the vertex-edge-cell interpolation of the solution u, and ‖⋅‖WG denotes the Weak Galerkin norm. The optimal uniform convergence order is demonstrated for semi-discrete and fully discrete schemes. Various numerical experiments are conducted to validate the optimal order of convergence demonstrated by the proposed method.

Improved Numerical Approach for Bagley–Torvik Equation Using Fractional Integral Formula and Adams–Moulton Method

Abstract In this paper, we introduce a numerical technique for solving Bagley–Torvik equations which plays an outstanding role in fractional calculus. To handle the derivatives and fractional integral in the Bagley–Torvik equations, the Laplace transform is employed to convert the equations to fractional integration equations. The resulting integral equations are solved by implicit Adams–Moulton methods. Moreover, we show the analytic convergence order of the proposed technique through the convergence analysis, and the analysis is validated by the numerical experiments. Illustrative experiments also demonstrate the validity and efficiency of the proposed method by comparing it with other existing methods.